Average Error: 7.3 → 0.4
Time: 1.0m
Precision: 64
\[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\mathsf{fma}\left(x - 1, \log y, \left(z - 1\right) \cdot \left(\log 1 - \mathsf{fma}\left(1, y, \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right)\right) - t\]
\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t
↓
\mathsf{fma}\left(x - 1, \log y, \left(z - 1\right) \cdot \left(\log 1 - \mathsf{fma}\left(1, y, \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right)\right) - t
double f(double x, double y, double z, double t) {
        double r154474 = x;
        double r154475 = 1.0;
        double r154476 = r154474 - r154475;
        double r154477 = y;
        double r154478 = log(r154477);
        double r154479 = r154476 * r154478;
        double r154480 = z;
        double r154481 = r154480 - r154475;
        double r154482 = r154475 - r154477;
        double r154483 = log(r154482);
        double r154484 = r154481 * r154483;
        double r154485 = r154479 + r154484;
        double r154486 = t;
        double r154487 = r154485 - r154486;
        return r154487;
}
↓
double f(double x, double y, double z, double t) {
        double r154488 = x;
        double r154489 = 1.0;
        double r154490 = r154488 - r154489;
        double r154491 = y;
        double r154492 = log(r154491);
        double r154493 = z;
        double r154494 = r154493 - r154489;
        double r154495 = log(r154489);
        double r154496 = 0.5;
        double r154497 = 2.0;
        double r154498 = pow(r154491, r154497);
        double r154499 = pow(r154489, r154497);
        double r154500 = r154498 / r154499;
        double r154501 = r154496 * r154500;
        double r154502 = fma(r154489, r154491, r154501);
        double r154503 = r154495 - r154502;
        double r154504 = r154494 * r154503;
        double r154505 = fma(r154490, r154492, r154504);
        double r154506 = t;
        double r154507 = r154505 - r154506;
        return r154507;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Derivation

  1. Initial program 7.3

    \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t\]

  2. Simplified7.3

    \[\leadsto \color{blue}{\mathsf{fma}\left(x - 1, \log y, \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t}\]

  3. Taylor expanded around 0 0.4

    \[\leadsto \mathsf{fma}\left(x - 1, \log y, \left(z - 1\right) \cdot \color{blue}{\left(\log 1 - \left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right)}\right) - t\]

  4. Simplified0.4

    \[\leadsto \mathsf{fma}\left(x - 1, \log y, \left(z - 1\right) \cdot \color{blue}{\left(\log 1 - \mathsf{fma}\left(1, y, \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right)}\right) - t\]

  5. Final simplification0.4

    \[\leadsto \mathsf{fma}\left(x - 1, \log y, \left(z - 1\right) \cdot \left(\log 1 - \mathsf{fma}\left(1, y, \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right)\right) - t\]

Reproduce

herbie shell --seed 2019322 +o rules:numerics
(FPCore (x y z t)
  :name "Statistics.Distribution.Beta:$cdensity from math-functions-0.1.5.2"
  :precision binary64
  (- (+ (* (- x 1) (log y)) (* (- z 1) (log (- 1 y)))) t))